wishart: Wishart Distribution
In Boom: Bayesian Object Oriented Modeling

wishart

R Documentation

Wishart Distribution

Description

Density and random generation for the Wishart distribution.

Usage

dWishart(W, Sigma, nu, logscale = FALSE)
rWishart(nu, scale.matrix, inverse = FALSE)

Arguments

`W`	Argument (random variable) for the Wishart density. A symmetric positive definite matrix.
`Sigma`	Scale or "variance" parameter of the Wishart distribution. See the "details" section below.
`nu`	The "degrees of freedom" parameter of the Wishart distribution. The distribution is only defined for `nu >= nrow(W) - 1`.
`logscale`	Logical. If `TRUE` then the density is returned on the log scale. Otherwise the density is returned on the density scale.
`scale.matrix`	For the Wishart distribution the `scale.matrix` parameter means the same thing as the `Sigma` parameter in `dWishart`. It is the variance parameter of the generating multivariate normal distribution. If simulating from the inverse Wishart, `scale.matrix` is the INVERSE of the "sum of squares" matrix portion of the multivariate normal sufficient statistics.
`inverse`	Logical. If TRUE then simulate from the inverse Wishart distribution. If FALSE then simulate from the Wishart distribution.

Details

If nu is an integer then a W(\Sigma, \nu) random variable can be thought of as the sum of nu outer products: y_iy_i^T, where y_i is a zero-mean multivariate normal with variance matrix Sigma.

The Wishart distribution is

\frac{|W|^{\frac{\nu - p - 1}{2}} \exp(-tr(\Sigma^{-1}W) / 2)}{ 2^{\frac{\nu p}{2}}|\Sigma|^{\frac{\nu}{2}}\Gamma_p(\nu / 2)}%

where p == nrow(W) and \Gamma_p(\nu) is the multivariate gamma function (see lmgamma).

Value

dWishart returns the density of the Wishart distribution. It is not vectorized, so only one random variable (matrix) can be evaluated at a time.

rWishart returns one or more draws from the Wishart or inverse Wishart distributions. If n > 0 the result is a 3-way array. Unlike the rWishart function from the stats package, the first index corresponds to draws. This is in keeping with the convention of other models from the Boom package.